This book, the third in a series of annual volumes, is an anthology of high-quality writing about various aspects of mathematics. Like its predecessors in 2011 and 2010, both of which have been reviewed in this column, the 2012 volume covers a number of topics in or about mathematics, ranging from the distribution of prime numbers to one person’s quest to find the perfect teaching method, with lots of stuff in between.
Two dozen articles are reprinted here. A number of them discuss substantive mathematical ideas: Terence Tao’s Structure and Randomness in the Prime Numbers, for example, addresses patterns in the set of prime numbers, and Robert Lang’s Flat-Unfoldability and Woven Origami Tesselations addresses the mathematical question of when origami patterns can be folded flat. Hayes’ An Adventure in the Nth Dimension provides some interesting observations about how the volume of a unit ball in n-space varies with n: it turns out to increase until we get to n = 5, and then starts to decrease from then on; what is special about n = 5?
There are also several articles that relate mathematics to other fields that are not generally thought of as scientific. Dancing Mathematics and the Mathematics of Dance by belcastro and Schaffer (both of whom are mathematicians and dancers), for example, talks about the connections between symmetry and group theory and dance. Schneiderman’s Can One Hear the Sound of a Theorem?, written by a person who is both a mathematician and musician, addresses the connections between those disciplines, and Mathematics Meets Photography: The Viewable Sphere by Swart and Torrence discusses the question of projecting a sphere onto a plane, with applications to photography; photographs by both authors appear in the article.
Though I haven’t read the earlier books in this series, I suspect, just from reading their reviews in this column, that the level of mathematics may be inching up. The review of the 2010 book, for example, commented on the fact that very little mathematical notation was used. In this volume, by contrast, I saw lots of graphs of functions, as well as integrals, set notation, and infinite sums and products (using the upper case sigma and pi notation), including the definition of the Riemann zeta function as an infinite sum. Nevertheless, this observation should not scare people away: the majority of articles in this book can be understood by a more general audience without specialized training.
Some of the articles, for example, are devoted to the teaching of mathematics. Illustrative of these is Erica Flapan’s light and amusing How to Be a Good Teacher is an Undecidable Problem, which recounts some of her experiences trying to find the “right” way to teach (“[l]et’s not even mention the trouble I got myself into trying to talk about the importance of balls in metric spaces”), all of which ultimately led her to the realization that there is no “right way” and everybody should just do what works for them — a refreshing change of pace from didactic articles by those who believe that their way of doing something is the only way. Other articles on mathematical pedagogy include How Your Philosophy of Mathematics Impacts Your Teaching by Bonnie Gold, the thesis of which is that an instructor’s attitude towards mathematics affects the way he or she teaches it, and the very short article by Mumford and Garfunkel, Bottom Line on Mathematics Education, which critiques the Common Core State Standards in Mathematics and argues for an approach to teaching that emphasizes the practical uses of mathematics.
The history and philosophy of mathematics are particularly well-represented in this volume. Taking the latter first, we have, for example, Gowers’ article Is Mathematics Discovered or Invented?, which addresses a well-known problem in mathematical philosophy, a problem that is also discussed in Livio’s article Why Math Works. The general theme of this latter article is why mathematics is so good at both predicting and explaining natural phenomena, a point that is also addressed in Peter Rowlett’s The Unplanned Impact of Mathematics, which essentially consists of seven short contributions from various members of the British Society for the History of Mathematics, each describing some surprising use of mathematical ideas (for example, Graham Hoare contributes a four-paragraph discussion of how relativity theory makes use of differential geometric ideas developed previously).
As for the history of mathematics, Fernando Gouvêa’s interesting Was Cantor Surprised? analyzes correspondence between Cantor and Dedekind to critically examine a story that “has grown in the telling” (namely the idea that Cantor himself did not believe the result that sets of different dimension can have the same cardinality); the analysis of this correspondence is not only interesting in its own right but also, in Gouvêa’s words, “reminds us of the importance of [social] interaction in the development of mathematics.” (This article, which originally appeared in the Monthly, is one of six articles in this anthology that first appeared in MAA journals; the number of such articles included in these volumes seems to be monotonically increasing from 2010 on.) Other historically-themed articles include Augustus de Morgan Behind the Scenes by Simmons, which looks at the role de Morgan played as mentor to other mathematicians, and Alexanderson’s The Cycloid and Jean Bernoulli, which recounts some history of this interesting curve, including the rivalry between the Bernoulli brothers about proving that it is the solution to the brachistochrone problem.
Some articles are a little harder to characterize. Baez’s The Strangest Numbers in String Theory provides not only a quick history of the quaternions and octonions but also comments on the role played by the latter in modern physics, and Mating, Dating and Mathematics: It’s All in the Game by Mark Colyvan takes a (non-technical) game-theoretic look at matters of the heart.
In addition to all this, two other features of the book should be noted. David Mumford has contributed a foreword which is itself an essay on the connections between pure and applied mathematics, and Pitici’s Introduction also contains a nice bibliography of interesting mathematics books published in the last year.
The articles in this anthology vary in length from 3 pages to around 20, with the average around 10, give or take a page or two. I finished some in about 15 minutes, while others required a greater investment of time; most could be comfortably read in an hour or so, however. The survey of articles that I have provided above is non-exhaustive; considerations of space make it impracticable for me to discuss every article (and I hasten to point out that the exclusion of any article says nothing about my opinion of its quality). I pretty much enjoyed all the articles here, and if the summary above whets your appetite for more, be sure to take a look at the book; odds are good that you’ll find something in it that strikes your fancy. As somebody who enjoys expository articles but generally doesn’t have the time to track them down and read them, finding a hand-picked collection like this assembled in one place was a delight. I look forward to seeing what 2013 brings.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.
Foreword: The Synergy of Pure and Applied Mathematics, of the Abstract and the Concrete
David Mumford ix
Mircea Pitici xvii
Why Math Works
Mario Livio 1
Is Mathematics Discovered or Invented?
Timothy Gowers 8
The Unplanned Impact of Mathematics
Peter Rowlett 21
An Adventure in the Nth Dimension
Brian Hayes 30
Structure and Randomness in the Prime Numbers
Terence Tao 43
The Strangest Numbers in String Theory
John C. Baez and John Huerta 50
Mathematics Meets Photography: The Viewable Sphere
David Swart and Bruce Torrence 61
Dancing Mathematics and the Mathematics of Dance
Sarah-Marie Belcastro and Karl Schaffer 79
Can One Hear the Sound of a Theorem?
Rob Schneiderman 93
Flat-Unfoldability and Woven Origami Tessellations
Robert J. Lang 113
A Continuous Path from High School Calculus to University Analysis
Timothy Gowers 129
Mathematics Teachers' Subtle, Complex Disciplinary Knowledge
Brent Davis 135
How to Be a Good Teacher Is an Undecidable Problem
Erica Flapan 141
How Your Philosophy of Mathematics Impacts Your Teaching
Bonnie Gold 149
Variables in Mathematics Education
Susanna S. Epp 163
Bottom Line on Mathematics Education
David Mumford and Sol Garfunkel 173
History of Mathematics and History of Science Reunited?
Jeremy Gray 176
Augustus De Morgan behind the Scenes
Charlotte Simmons 186
Routing Problems: A Historical Perspective
Giuseppe Bruno, Andrea Genovese, and Gennaro Improta 197
The Cycloid and Jean Bernoulli
Gerald L. Alexanderson 209
Was Cantor Surprised?
Fernando Q. Gouvêa 216
Why Is There Philosophy of Mathematics at All?
Ian Hacking 234
Ultimate Logic: To Infinity and Beyond
Richard Elwes 255
Mating, Dating, and Mathematics: It's All in the Game
Mark Colyvan 262
Notable Texts 281